We study the inclusive K--meson production in proton-nucleus collisions in the subthreshold energy regime in the framework of a spectral function approach for incoherent primary proton-nucleon and secondary pion-nucleon production processes. Our approach takes properly into account the effect of the nuclear mean-field potentials on these processes, as well as the final-state interaction (FSI) among the outgoing nucleons participating in the one-step antikaon creation process. A detailed comparison of the model calculations of the K- differential cross-sections for the reactions p + Be-9, P + Cu-63, and p + Au-197 at subthreshold energies with the currently available experimental data obtained recently at the ITEP proton synchrotron and at SIS/GSI is given. We find that our calculation, which includes both the nuclear density-dependent mean-field potentials and the elementary NN-FSI. effects,on the K- production from the direct mechanism, is able to reproduce the energy dependences of the invariant differential cross-sections for the "hard" antikaon creation in p-Be-9 and P-Cu-63 collisions. This result contradicts previous estimates which used only density-dependent mean fields in calculating the K- yield from this mechanism. We further show that the NN-FSI effects play a minor role in describing the data on the spectrum of the relatively soft K--mesons from p-(197) An interactions at incident energy of 2.5 GeV. We find that the relative strength of the proton- and pion-induced reaction channels in the subthreshold energy regime is governed by the kinematics of the experiment under consideration. We also explore the influence of the antikaon mean-field potential on the K- yield at low antikaon momenta.